How can make my puffy nipples go away? (Photos)

The tissue below the areola area is pushing it outward.  This is easily corrected with surgery through a distant incision on the outside rib cage and using long instruments.  All the best, Dr. Joe

From your photos you appear to have gynecomastia (which would need to be verified through Skype or an in-office consult), the cause for which is frequently idiopathic, meaning that the cause is unknown in most cases. There are numerous other causes for gynecomastia including anabolic steroids, marijuana, certain prescription medications, liver failure, and certain tumors, among other things.  Smaller gynecomastia procedures are performed under local anesthesia and oral sedation.  This is a very well tolerated procedure that takes about an hour or less to complete.  The patient goes home the same day and begins the recovery period.
The larger gynecomastia correction procedure which involves greater tissue removal as well as skin removal is usually performed under general anesthesia.  This male reduction procedure takes less than 2 hours to complete, even in the most complicated of cases. The patient goes home the same day.